ECE 6382 Fall 2019 David R Jackson Notes
- Slides: 26
ECE 6382 Fall 2019 David R. Jackson Notes 2 Differentiation of Functions of a Complex Variable Notes are adapted from D. R. Wilton, Dept. of ECE 1
Functions of a Complex Variable 2
Differentiation of Functions of a Complex Variable 3
The Cauchy – Riemann Conditions Denote Augustin-Louis Cauchy Bernhard Riemann 4
The Cauchy – Riemann Conditions (cont. ) Cauchy-Riemann equations 5
The Cauchy – Riemann Conditions (cont. ) Arbitrary direction z 6
The Cauchy – Riemann Conditions (cont. ) Hence we have the following equivalent statements: 7
The Cauchy – Riemann Conditions (cont. ) D 8
Applying the Cauchy – Riemann Conditions 9
Applying the Cauchy – Riemann Conditions (cont. ) D A singularity is a point where the function is not analytic. 10
Applying the Cauchy – Riemann Conditions (cont. ) 11
Differentiation Rules (cont. ) Note: The above “brute-force” derivation, directly using the definition of the derivative, is exactly what is done is usual calculus, with x being used there instead of z. 12
Differentiation Rules 13
Differentiation Rules 14
A Theorem Related to z* If f = f (z, z*) is analytic, then (The function cannot really vary with z*, and therefore cannot really be a function of z* except in a trivial way. ) 15
A Theorem Related to z* (cont. ) 16
A Theorem Related to z* (cont. ) 17
Entire Functions A function that is analytic everywhere in the finite* complex plane is called “entire”. * A function is said to be analytic everywhere in the finite complex plane if it is analytic everywhere except possibly at infinity. Analytic at infinity: Is the function analytic at w = 0? 18
Combinations of Analytic Functions Combinations of functions: 19
Combinations of Analytic Functions (cont. ) Combinations of functions: Example: The first form is analytic everywhere except z = 1. The second form is analytic for |z| < 1. 20
Combination of Analytic Functions (cont. ) Examples Composite functions of analytic functions are also analytic Derivatives of analytic functions are also analytic 21
Derivatives of Analytic Function Important theorem (proven later) The derivative of an analytic function is also analytic. Hence, all derivatives of an analytic function are also analytic. 22
Real and Imaginary Parts of Analytic Functions Are Harmonic Functions The functions u and v are harmonic (i. e. , they satisfy Laplace’s equation) Notation: This result is extensively used in conformal mapping to solve electrostatics and other problems involving the 2 D Laplace equation (discussed later). Pierre-Simon Laplace 23
Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Proof f is analytic df / dz is also analytic (see note on slide 22) 24
Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Example: 25
Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Example: 26
- Andrew jackson doodle
- David r jackson
- David r. jackson
- Facteur g
- Michael jackson x ray
- Andrew jackson movie
- David schwimmer mbti
- What did the sibley commission do
- The lottery shirley jackson questions
- Walter jackson freeman iii
- General jackson slaying the many headed monster
- Michael jackson personality
- Andrew jackson trail of tears map
- The possibility of evil questions
- The lottery shirley jackson questions
- Why can’t percy simply fight crusty?
- Who is the “familiar face” that percy thinks he sees?
- Lightning thief chapter 12 summary
- What does aunty em look like when she opens the door?
- The lightning thief chapter 6 pdf
- Chapter 2 percy jackson the lightning thief
- Intersurgical jackson rees
- Peeples middle school
- Sa sd
- Shirley jackson seven types of ambiguity
- Loulou lemmon
- Clase iv de kennedy diseño